Combining a smooth information criterion with neural networks

# Combining a smooth information criterion with neural networks
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## **Andrew McInerney**, ** **
### University of Limerick
#### LMU, 07 July 2023

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# Where is Limerick?

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# Limerick

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# Univeristy of Limerick

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# Background

*  Research: Neural networks from a statistical-modelling perspective

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class: selectnn-slide
# Model Selection

A Statistically-Based Approach to Feedforward Neural Network Model Selection (arXiv:2207.04248)

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class: selectnn-slide
# Insurance: Model Selection

```r
library(selectnn)
nn <- selectnn(charges ~ ., data = insurance, Q = 8,
               n_init = 5)
summary(nn)
```

```{.bg-primary}
## [...]
## Number of input nodes: 4 
## Number of hidden nodes: 2 
## 
## Value: 1218.738 
##         Covariate Selected Delta.BIC
##        smoker.yes      Yes  2474.478
##               bmi      Yes   919.500
##               age      Yes   689.396
##          children      Yes    13.702
## [...]
```

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Extend packages: **nnet**, **neuralnet**, **keras**, **torch**

*  Significance testing

*  Covariate-effect plots

---

```r
intnn <- interpretnn(nn)  
summary(intnn)
```

```{.bg-primary}
## Coefficients:
##                                Weights |      X^2    Pr(> X^2)
##               age     (-0.43***, 0.04) |  41.4363 1.01e-09 ***
##          sex.male        (0.08*, 0.13) |   5.5055 6.38e-02 .  
##               bmi      (0.03, 2.19***) | 105.6106 0.00e+00 ***
##          children   (-0.08***, -0.11.) |  19.0146 7.43e-05 ***
##        smoker.yes (-3.16***, -6.19***) | 250.6393 0.00e+00 ***
##  region.northwest        (0.07., 0.15) |   2.8437 2.41e-01    
##  region.southeast        (0.11*, 0.12) |   6.2560 4.38e-02 *  
##  region.southwest       (0.15**, 0.14) |  10.8218 4.47e-03 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
```

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```r
plotnn(intnn)
```

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```r
plot(intnn, conf_int = TRUE, which = c(1, 4))
```

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# Current Work

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# Smooth Information Criterion

$$
 \text{IC} = -2\ell(\theta) + \lambda [\lVert \tilde\theta \rVert_0 + 1]
$$

where `$\lambda =  \log(n)$` for the BIC.

Rearrange as an IC-based penalized likelihood:

`$$\ell^{\text{IC}}(\theta) = \ell(\theta) - \frac{\log(n)}{2} [\lVert \tilde\theta \rVert_{0} + 1]$$`

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# Smooth Information Criterion

Introduce "smooth `$L_0$` norm":

`$$\lVert \theta \rVert_{0, \epsilon} = \sum_{j=1}^p \phi_\epsilon (\theta_j)$$`

where

$$
 \phi_\epsilon(\theta_j) = \frac{{\theta_j^2}}{\theta_j^2 + \epsilon^2}
$$

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# Smooth Information Criterion

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# Motivation

* Tuning parameter automatically selected in one step

<br>

* Computationally advantageous

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# `$\epsilon$`-telescoping

* Optimal `$\epsilon$` is zero

* Smaller `$\epsilon$` `$\implies$` less numerically stable

* Start with larger `$\epsilon$`, and "telescope" through a decreasing sequence of `$\epsilon$` values using warm starts

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# R Package

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# Extending to Neural Networks

`$$\mathbb{E}(y) = \text{NN}(X, \theta)$$`

where

`$$\text{NN}(X, \theta) = \phi_o \left[ \gamma_0+\sum_{k=1}^q \gamma_k \phi_h \left( \sum_{j=0}^p \omega_{jk}x_{j}\right) \right]$$`

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# Extending to Neural Networks

We can then formulate a **smooth** BIC-based penalized likelihood:

`\begin{equation*}
  \ell^{\text{SIC}}(\theta) = \ell(\theta) - \frac{\log(n)}{2} [\lVert \tilde\theta \rVert_{0, \epsilon} + q + 1],
\end{equation*}`

--
where
`\begin{equation*}
  \ell(\theta)= -\frac{n}{2}\log(2\pi\sigma^2)-\frac{1}{2\sigma^2}\sum_{i=1}^n(y_i-\text{NN}(x_i))^2
\end{equation*}`

---
# Extending to Group Sparsity

The smooth approximation of the `$L_0$` norm can be written for groups as

$$
 \text{card}(\theta) \times \phi_\epsilon(||\theta||_2^2) = \text{card}(\theta) \times \frac{||\theta||_2^2}{||\theta||_2^2 + \epsilon^2}. 
$$

---
class: inputgroup-slide
# Group Sparisty

## Input-neuron penalization

<font size="6">
`\begin{equation*}
  \ell^{\text{IN-SIC}}(\theta) = \ell(\theta) - \frac{\log(n)}{2} \left[q \times \sum_{j=1}^p\big\Vert\Vert\omega_j\Vert_2^2\big\Vert_{0,\epsilon}  + \lVert \tilde\gamma \rVert_{0, \epsilon} + q + 1\right]
\end{equation*}`
</font>
where `$\omega_{j} = (\omega_{j1},\omega_{j2},\dotsc,\omega_{jq})^T$`

---
class: hiddengroup-slide
# Group Sparisty

## Hidden-neuron penalization

<font size="6">
`\begin{equation*}
  \ell^{\text{HN-SIC}}(\theta) = \ell(\theta) - \frac{\log(n)}{2} \left[(p+1) \times \sum_{k=1}^q\big\Vert\Vert\theta^{(k)}\Vert_2^2\big\Vert_{0,\epsilon} + q + 1\right]
\end{equation*}`
</font>
where `$\theta^{(k)} = (\omega_{1k},\omega_{2k},\dotsc,\omega_{pk}, \gamma_k)^T$`

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# Combined Penalty

* Implement a group penalty and the single-parameter penalty in one optimization procedure

* Start with group penalization and telescope through the `$\epsilon$` values until some predefined value, `$\tau$`

* Switch to single-parameter penalization for the remainder of the `$\epsilon$` values

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# Approaches

* Single-parameter penalization

* Input-neuron penalization

* Hidden-neuron penalization

* Combined approaches (perform group penalization initially and then switch to single-parameter penalization)

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# Preliminary Simulation

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# Preliminary Results

<img src="data:image/png;base64,#img/prelim-results-extra.png" width="90%" style="display: block; margin: auto;" />
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class: bigger
# References

*  <font size="5">McInerney, A., & Burke, K. (2022). A statistically-based approach to feedforward neural network model selection. <i>arXiv preprint arXiv:2207.04248</i>.  </font>

*  <font size="5">McInerney, A., & Burke, K. (2023). Interpreting feedforward neural networks as statistical models. <i>To appear on arXiv</i>.    </font>

*  <font size="5">O’Neill, M. and Burke, K. (2023). Variable selection using a smooth information criterion for distributional regression models. <i>Statistics and Computing, 33(3), p.71</i>.    </font>

### R Packages

```r
devtools::install_github(c("andrew-mcinerney/selectnn",
                           "andrew-mcinerney/interpretnn"))
```